### Abstract

The problem of computing the fuzzy weighted average, where both attributes and weights are fuzzy numbers, is well studied in the literature. Generally, the approach is to apply Zadeh’s extension principle to compute α-cuts of the fuzzy weighted average from the α-cuts of the attributes and weights for fixed values of α∈[0..1]; this means that all values of the membership functions of the fuzzy weighted average are computed separately. In this paper, we generalise this approach in such a way that α is considered to be a parameter; this enables us to compute exact analytical membership functions for the fuzzy weighted average in case the attributes and weights are triangular or trapeizoidal fuzzy numbers. To illustrate the power of our algorithms, they are applied to the examples from the literature, providing exact membership functions in each case.

Original language | Undefined |
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Title of host publication | Computational Intelligence |

Place of Publication | Berlin / Heidelberg |

Publisher | Springer |

Pages | 85-99 |

Number of pages | 15 |

ISBN (Print) | 978-3-642-20206-3 |

DOIs | |

Publication status | Published - 2011 |

### Publication series

Name | Studies in Computational Intelligence |
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Publisher | Springer |

Number | 343 |

Volume | 343 |

ISSN (Print) | 1860-949X |

ISSN (Electronic) | 1860-9503 |

### Keywords

- METIS-278714
- IR-77052
- EWI-20154
- Fuzzy weighted average
- membership functions

## Cite this

van den Broek, P. M., & Noppen, J. A. R. (2011). Exact Membership Functions for the Fuzzy Weighted Average. In

*Computational Intelligence*(pp. 85-99). (Studies in Computational Intelligence; Vol. 343, No. 343). Berlin / Heidelberg: Springer. https://doi.org/10.1007/978-3-642-20206-3_6